IAA-AAS-DyCoSS1 -08-06 DETECTING INVARIANT MANIFOLDS USING HYPERBOLIC LAGRANGIAN COHERENT STRUCTURES Daniel P ´ erez * , Gerard G ´ omez † , Josep J. Masdemont ‡ Using as reference test model the Planar Circular Restricted Three Body Prob- lem, this paper explores its Lagrangian Coherent Structures, as well as its Hy- perbolic Lagrangian Coherent Structures. The purpose is to identify stable and unstable manifolds acting as separatrices between orbits with different qualitative behaviour and, therefore, relevant to the dynamics of the problem. Particular at- tention is given to the manifolds associated to the collinear libration points and to the practical stability regions around the triangular equilibrium points. INTRODUCTION Lagrangian Coherent Structures (LCS), introduced by G. Haller et al. 5 for the study of dynamical systems, give a methodology to identify the boundaries between regions in the configuration space with orbits that have different dynamical behaviour. LCS are usually computed by means of Finite Time Lyapunov Exponents (FTLE) at time T , just looking at the FTLE scalar-field. The value of the FTLE at ~x gives an idea of the behaviour of the orbits around ~x: if the FTLE is small then the orbits in a neighbourhood of ~x will be close at time T ; however, if FTLE is high then the image under the flow, up to t = T , of points close to ~x will have different behaviours and usually some of them will tend to depart from the others. In particular a high value of a FTLE can be an indicator of the existence of an invariant manifold. According to Haller, 5 LCSs are defined as follows. Let D⊂ R n be an open set and f the vector field: f : D× R -→ R n (~x,t) 7-→ f (~x,t). Given a dynamical system defined by the ordinary differential equation: ˙ ~x(t)= f (~x(t),t), whose associated flow will be denoted by φ φ : R × R ×D -→ D (t, t 0 ,~ x 0 ) 7-→ φ(t; t 0 ,~x 0 ), * Departament de Matem` atica Aplicada i An` alisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain † IEEC & Departament de Matem` atica Aplicada i An` alisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain ‡ IEEC & Departament de Matem` atica Aplicada I, ETSEIB, Universitat Polit` ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain 1